Turbulence, its community, and our approach

Even if you have not studied turbulence, you already know a lot about it. You have seen the chaotic, ever-changing, three-dimensional nature of chimney plumes and flowing streams. You know that turbulence is a good mixer. You might have come across an article that described the intrigue it holds for mathematicians and physicists.

Unless a fluid flow has a low Reynolds number or very stable stratification (less dense fluid over more dense fluid), it is turbulent. Most flows in engineering, in the lower atmosphere, and in the upper ocean are turbulent. Because of its “mathematical intractability” – turbulence does not yield exact mathematical solutions – its study has always involved observations. But over the past three decades numerical approaches have proliferated; today they are a dominant means of studying turbulent flows.

Turbulence has long been studied in both engineering and geophysics. G. I. Taylor's contributions spanned both (Batchelor, 1996). The Lumley and Panofsky (1964) work was my introduction to that breadth, but as Lumley later commented, their parts of that text “just…touch.” Today the turbulence field seems more coherent than it was in 1964, although it still has subcommunities and dialects (Lumley and Yaglom, 2001).

In Part I of this book we focus on the physical understanding of turbulence, surveying its key properties. We'll use its governing equations to guide our discussions and inferences. We shall also discuss the main types of numerical approaches to turbulence.

Introduction and background

We saw in Chapter 1 that direct numerical solutions of the turbulence equations can be done only at relatively small values of the turbulence Reynolds number Rt. Calculations of the vastly larger Rt flows in engineering and geophysical applications use averaged forms of these equations. The averaging produces important turbulent fluxes that must be specified before the equations can be solved numerically. That specification allows today's calculations of averaged turbulence fields in applications ranging from flow in the atmospheric boundary layer to the general circulation of the earth's atmosphere and convection on the sun.

In this chapter we derive, interpret, and scale the conservation equations for several covariances, including turbulent fluxes, that arise from ensemble averaging. At very coarse resolution ergodicity (the property that any unbiased average converges to the ensemble average, Chapter 2) blurs the distinction between ensemble and space averaging, so these covariance equations are used also in traditional mesoscale-modeling and weather-forecasting applications.

Monin and Yaglom (1971) credit a 1924 paper by Keller and Friedmann as the first to present a method of deriving turbulence moment equations, of which the turbulent-flux budgets are an example. Because they contain unknown terms, the turbulence moment equations were little used until the late 1960s, when computers were large enough to solve approximate versions of them.

The literature on models of these flux budgets can be bewildering. The rationale for their closure approximations is not always discussed, and the models go under several names – e.g., “second-order closure,” “Reynolds-averaged Navier–Stokes (RANS),” “single-point closure,” “higher-order closure,” and “invariant modeling.”

Average and instantaneous properties contrasted

Figure 2.1 is a famous snapshot of a turbulent wake, the region downstream of a body in a moving fluid. The instantaneously thin, irregular boundary between the turbulent and nonturbulent flow is continuously deformed by the turbulent eddies, so that under averaging it becomes a broad, smooth transition region. Figure 2.2 illustrates this same feature at the top of the atmospheric boundary layer.

We have had access to instantaneous turbulence fields through remote sensing and numerical simulation only since the 1970s. Perhaps that is why our descriptive terms for turbulence tend to refer to its statistical properties, not its instantaneous ones. For example,

• Homogeneous turbulence has spatially uniform statistical properties (with the exception of mean pressure). A turbulent flow can be homogeneous in zero, one, two, or three directions. A sphere wake is an example of the first. The turbulent boundary layer near the leading edge of a flat plate, or downstream of a change in surface conditions, can be homogeneous in one direction (the lateral) but is inhomogeneous in the wall-normal and streamwise directions. The turbulent boundary layer over a uniform surface can be homogeneous in two directions, those in the plane parallel to the surface, but is necessarily inhomogeneous in the normal direction. The grid turbulence produced by a grating of bars spanning the cross section of a wind tunnel is homogeneous in the cross-stream plane but inhomogeneous in the streamwise direction because it decays as it goes downstream (Chapter 5). […]

Introduction

The stable boundary layer (SBL) is as different from the convective boundary layer (CBL) as night is from day. The SBL is typically much thinner and much less diffusive; chimney plumes in the stable air just after sunrise can travel intact for long distances, quite unlike those in midday. The decrease in surface wind speed around sunset on a clear day, which is prominent in wind climatologies (Arya, 2001), is also evident to casual observers. Figure 12.1 shows the root cause: in a given mean horizontal pressure gradient the surface mean wind speed is lower in stable stratification than in neutral or unstable stratification. Thus the lower nocturnal wind speeds tend to persist until the initiation of a CBL at sunrise.

We gave examples of SBLs in Chapter 9. The two sketched in Figure 9.7 are made stably stratified by flow over a cooler surface and by entrainment of warmer air aloft. Perhaps the most common example is the nocturnal SBL over land in fair weather. Another is the “long-lived” SBL at the South Pole; it is caused by a combination of a temperature inversion over a cooled, sloped surface and the typically downslope orientation of the mean pressure gradient (Neff et al., 2008).

As we'll see, turbulence in the SBL tends to be in a delicate dynamical balance, and so SBL structure tends to be more difficult to study and to parameterize than that in the CBL.

The spray equations have been studied and solved for many applications: single-component and multicomponent liquids, high-temperature and low-temperature gas environments, monodisperse and polydisperse droplet-size distributions, steady and unsteady flows, one-dimensional and multidimensional flows, laminar and turbulent regimes, subcritical and supercritical thermodynamic regimes, and recirculating (strongly elliptical) and nonrecirculating (hyperbolic, parabolic, or weakly elliptic) flows. The analyses discussed here will not be totally inclusive of all of the interesting analyses that have been performed; rather, only a selection is presented.

Spray flows can be classified in various ways. One important issue concerns whether the gas is turbulent or laminar. In this chapter, only laminar flows are considered; the turbulent situation is discussed in Chapter 10. Another issue concerns whether thermodynamic conditions are subcritical, on the one hand, or near critical to supercritical, on the other hand.

In the most general spray case, the gas and the droplets are not in thermal and kinematic equilibria, that is, the droplet temperature and the droplet velocity differ from those properties of the surrounding gas. Of course, heat transfer and drag forces result in the tendency to move toward equilibrium. The equilibrium case is sometimes described as a locally homogeneous flow. It is possible to have thermal equilibrium or kinematic equilibrium without the other. When thermal equilibrium exists, the analysis described in Chapter 7 is simplified because the droplet temperature Tl can be set equal to the gas temperature T and Eq. (7.82) or its alternative forms, Eq. (7.83) or Eq. (7.87), can be avoided.

The fluid dynamics and transport of sprays comprise an exciting field of broad importance. There are many interesting applications of spray theory related to energy and power, propulsion, heat exchange, and materials processing. Spray phenomena also have natural occurrences. Spray and droplet behaviors have a strong impact on vital economic and military issues. Examples include the diesel engine and gas-turbine engine for automotive, power-generation, and aerospace applications. Manufacturing technologies including droplet-based net form processing, coating, and painting are important applications. Applications involving medication, pesticides and insecticides, and other consumer uses add to the impressive list of important industries that use spray and droplet technologies. These industries involve annual production certainly measured in tens of billions of dollars and possibly higher. Many applications are still under development. The potentials for improved performance, improved market shares, reduced costs, and new products and applications are immense. Continuing effort is needed to optimize the designs of spray and droplet applications and to develop strategies and technologies for active control of sprays in order to achieve the huge potential.

In the first edition of this book and in this second edition, I have attempted to provide some scientific foundation for movement toward the goals of optimal design and effective application of active controls. The book, however, does not focus on design and controls. Rather, I discuss the fluid mechanics and transport phenomena that govern the behavior of sprays and droplets in the many important applications.

There are various complications that occur when a multicomponent liquid is considered (Landis and Mills, 1974, and Sirignano and Law, 1978). Different components vaporize at different rates, creating concentration gradients in the liquid phase and causing liquid-phase mass diffusion. The theory requires the coupled solutions of liquid-phase species-continuity equations, multicomponent phase-equilibrium relations (typically Raoult's Law), and gas-phase multicomponent energy and species-continuity equations. Liquid-phase mass diffusion is commonly much slower than liquid-phase heat diffusion so that thin diffusion layers can occur near the surface, especially at high ambient temperatures at which the surface-regression rate is large. The more volatile substances tend to vaporize faster at first until their surface-concentration values are diminished and further vaporization of those quantities becomes liquid-phase mass diffusion controlled.

Mass diffusion in the liquid phase is very slow compared with heat diffusion in the liquid and extremely slow compared with momentum, heat, or mass diffusion in the gas film or compared with momentum diffusion in the liquid. In fact, the characteristic time for the liquid-phase mass diffusion based on droplet radius is typically longer than the droplet lifetime. Nevertheless, this mass diffusion is of primary importance in the vaporization process for a multicomponent fuel. At first, early in the droplet lifetime, the more volatile substances in the fuel at the droplet surface will vaporize, leaving only the less volatile material that vaporizes more slowly. More volatile material still exists in the droplet interior and will tend to diffuse toward the surface because of concentration gradients created by the prior vaporization.

Consider the gas phase that surrounds or adjoins liquid droplets, films, pools, and/or streams. Figure B.1 presents a diagram that portrays a wide variety of two-phase reacting flow situations included here. Laminar multicomponent flow with viscosity, Fourier heat conduction, and Fickian mass diffusion will be analyzed. Diffusivities for all species will be assumed identical; the Lewis number value will be unity; radiation will be neglected; and kinetic energy will be neglected in comparison with thermal energy so that, with regard to the energetics, pressure will be considered uniform over the space although the pressure gradient can be significant in the momentum balance (small Mach number). We will analyze, as a general case, the situation in which a one-step exothermic reaction occurs between the ambient gas and the vapor from the bulk liquid. We will refer to the liquid as the fuel and the ambient gas as the oxidizer, although the opposite situation, e.g., oxygen droplet in hydrogen gas, can be analyzed in exactly the same fashion. The nonreacting case can be the special subcase in which the reaction rate becomes zero.

Both steady and unsteady scenarios will be discussed. The roles of the gas-phase energy equation and the species equations will be emphasized. Of course, they must be coupled with the continuity and momentum equations and with a gas equation of state. In addition, the solution of conservation equations for the liquid phase might be required.

There is interest in the droplet-vaporization problem from two different aspects. First, we wish to understand the fluid-dynamic and -transport phenomena associated with the transient heating and vaporization of a droplet. Second, but just as important, we must develop models for droplet heating, vaporization, and acceleration that are sufficiently accurate and simple to use in a spray analysis involving so many droplets that each droplet's behavior cannot be distinguished; rather, an average behavior of droplets in a vicinity is described. We can meet the first goal by examining both approximate analyses and finite-difference analyses of the governing Navier–Stokes equations. The second goal can be addressed at this time with only approximate analyses because the Navier–Stokes resolution for the detailed flow field around each droplet is too costly in a practical spray problem. However, correlations from Navier–Stokes solutions provide useful inputs into approximate analyses. The models discussed herein apply to droplet vaporization, heating, and acceleration and to droplet condensation, cooling, and deceleration for a droplet isolated from other droplets. The governing partial differential equations reflecting the conservation laws are presented in Appendix A. Several coordinate systems are considered. Formulations with primitive velocity variables and formulations with stream functions are discussed. Appendix B discusses some conserved scalar variables whose analytical use can be very convenient and powerful under certain ideal conditions.

Introductory descriptions of vaporizing droplet behavior can be found in the works of Chigier (1981), Clift et al. (1978), Glassman (1987), Kanury (1975), Kuo (1986), Lefebvre (1989), and Williams (1985).